$( 1 + i ) ( 2 - i ) =$
A. $- 3 - \mathrm { i }$
B. $- 3 + \mathrm { i }$
C. $3 - i$
D. $3 + i$

Given the complex number $z = 2 + \mathrm { i }$, then $z \cdot \bar { z } =$ (A) $\sqrt { 3 }$ (B) $\sqrt { 5 }$ (C) 3 (D) 5

Let $z = - 3 + 2 i$. Then in the complex plane, the point corresponding to $\bar { z }$ is located in
A. the first quadrant
B. the second quadrant
C. the third quadrant
D. the fourth quadrant

If $z ( 1 + \mathrm { i } ) = 2 \mathrm { i }$ , then $z =$
A. $- 1 - \mathrm { i }$
B. $- 1 + \mathrm { i }$
C. $1 - \mathrm { i }$
D. $1 + \mathrm { i }$

3. The main content of this test paper covers all content of the college entrance examination.

Section I

I. Multiple Choice Questions: This section contains 12 questions, each worth 5 points, totaling 60 points. For each question, only one of the four options is correct.
1. The conjugate of the complex number $z = \mathrm { i } ^ { 9 } ( - 1 - 2 \mathrm { i } )$ is
A. $2 + \mathrm { i }$
B. $2 - \mathrm { i }$
C. $- 2 + \mathrm { i }$
D. $- 2 - \mathrm { i }$
2. Let sets $A = \{ a , a + 1 \} , ~ B = \{ 1,2,3 \}$. If $A \cup B$ has 4 elements, then the set of possible values of $a$ is
A. $\{ 0 \}$
B. $\{ 0,3 \}$
C. $\{ 0,1,3 \}$
D. $\{ 1,2,3 \}$
3. For the hyperbola $C : \frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > 0 , b > 0 )$, the length of the real axis and the focal distance are 2 and 4 respectively. The asymptote equations of hyperbola $C$ are
A. $y = \pm \frac { \sqrt { 3 } } { 3 } x$
B. $y = \pm \frac { 1 } { 3 } x$ C. $y = \pm \sqrt { 3 } x$
D. $y = \pm 3 x$

If $z = 1 + \mathrm { i }$, then $\left| z ^ { 2 } - 2 z \right| =$
A. 0
B. 1
C. $\sqrt { 2 }$
D. 2

If $z = 1 + 2 \mathrm { i } + \mathrm { i } ^ { 3 }$ , then $| z | =$
A. 0
B. 1
C. $\sqrt { 2 }$
D. 2

If $\bar { z } ( 1 + \mathrm { i } ) = 1 - \mathrm { i }$, then $z =$
A. $1 - \mathrm { i }$
B. $1 + \mathrm { i }$
C. $- i$
D. $i$

The imaginary part of the complex number $\frac { 1 } { 1 - 3 \mathrm { i } }$ is
A. $- \frac { 3 } { 10 }$
B. $- \frac { 1 } { 10 }$
C. $\frac { 1 } { 10 }$
D. $\frac { 3 } { 10 }$

Given that the complex number $z$ satisfies $z = 1 - 2 i$ ($i$ is the imaginary unit), find $| z | =$ $\_\_\_\_$

1. The point corresponding to the complex number $\frac { 2 - \mathrm { i } } { 1 - 3 \mathrm { i } }$ in the complex plane is located in which quadrant?
A. First quadrant
B. Second quadrant
C. Third quadrant
D. Fourth quadrant 【Answer】A 【Solution】 【Analysis】Use complex division to simplify $\frac { 2 - \mathrm { i } } { 1 - 3 \mathrm { i } }$, and thus determine the location of the corresponding point. 【Detailed Solution】 $\frac { 2 - \mathrm { i } } { 1 - 3 \mathrm { i } } = \frac { ( 2 - \mathrm { i } ) ( 1 + 3 \mathrm { i } ) } { 10 } = \frac { 5 + 5 \mathrm { i } } { 10 } = \frac { 1 + \mathrm { i } } { 2 }$, so the point corresponding to this complex number is $\left( \frac { 1 } { 2 } , \frac { 1 } { 2 } \right)$, which is located in the first quadrant. Therefore, the answer is: A.

2. C
Solution: $z ( \bar { z } - i ) = ( 2 - i ) ( 2 + 2 i ) = 6 + 2 i$, so the answer is $C$.

3. Given $( 1 - i ) ^ { 2 } z = 3 + 2 i$, then $z =$
A. $- 1 - \frac { 3 } { 2 } i$
B. $- 1 + \frac { 3 } { 2 } i$
C. $- \frac { 3 } { 2 } + i$
D. $- \frac { 3 } { 2 } - i$

3. Given $(1 - i)^2 z = 3 + 2i$, then $z =$
A. $-1 - \frac{3}{2}i$
B. $-1 + \frac{3}{2}i$
C. $-\frac{3}{2} + i$
D. $-\frac{3}{2} - i$

16. $5240 \left( 3 - \frac { n + 3 } { 2 ^ { n } } \right)$
Solution: According to the pattern, for a given $n$, folding $n$ times produces figures with dimensions of the form $\left( \frac { 20 } { 2 ^ { k } } \right) \text{ dm} \times \left( \frac { 10 } { 2 ^ { k } } \right) \text{ dm}$ for $k = 0, 1, \cdots, n$. The number of different sizes is $n + 1$. When $n = 4$, there are 5 different sizes. The area of each size is $S _ { n } = \frac { 240 ( n + 1 ) } { 2 ^ { n } }$. Therefore,
$$\begin{gathered} \sum _ { k = 1 } ^ { n } S _ { k } = 240 \sum _ { k = 1 } ^ { n } \frac { k + 1 } { 2 ^ { k } } = 240 \left( 2 \sum _ { k = 1 } ^ { n } \frac { k + 1 } { 2 ^ { k } } - \sum _ { k = 1 } ^ { n } \frac { k + 1 } { 2 ^ { k } } \right) \\ = 240 \left( \sum _ { k = 0 } ^ { n - 1 } \frac { k + 2 } { 2 ^ { k } } - \sum _ { k = 1 } ^ { n } \frac { k + 1 } { 2 ^ { k } } \right) = 240 \left( 2 - \frac { n + 1 } { 2 ^ { n } } + \sum _ { k = 1 } ^ { n - 1 } \frac { 1 } { 2 ^ { k } } \right) \\ = 240 \left( 3 - \frac { n + 3 } { 2 ^ { n } } \right) \left( \text{dm} ^ { 2 } \right) \end{gathered}$$

IV. Solution Questions

If $z = - 1 + \sqrt { 3 } \mathrm { i }$ , then $\frac { z } { z \bar { z } - 1 } =$
A. $- 1 + \sqrt { 3 } \mathrm { i }$
B. $- 1 - \sqrt { 3 } \mathrm { i }$
C. $- \frac { 1 } { 3 } + \frac { \sqrt { 3 } } { 3 } \mathrm { i }$
D. $- \frac { 1 } { 3 } - \frac { \sqrt { 3 } } { 3 } \mathrm { i }$

The point corresponding to the complex number $\frac { 2 - \mathrm { i } } { 1 - 3 \mathrm { i } }$ in the complex plane is located in which quadrant?
A. First quadrant
B. Second quadrant
C. Third quadrant
D. Fourth quadrant

Let $( 1 + 2 \mathrm { i } ) a + b = 2 \mathrm { i }$ , where $a , b$ are real numbers, then
A. $a = 1 , b = - 1$
B. $a = 1 , b = 1$
C. $a = - 1 , b = 1$
D. $a = - 1 , b = - 1$

Given $z = 1 - 2i$, and $z + a\bar{z} + b = 0$, where $a, b$ are real numbers, then
A. $a = 1, b = -2$
B. $a = -1, b = 2$
C. $a = 1, b = 2$
D. $a = -1, b = -2$

2. If $\mathrm { i } ( 1 - z ) = 1$, then $z + \bar { z } =$
A. $- 2$
B. $- 1$
C. $1$
D. $2$

Let $z = \frac { 2 + i } { 1 + i ^ { 2 } + i ^ { 5 } }$, then $\bar { z } =$
A. $1 - 2 i$
B. $1 + 2 i$
C. $2 - i$
D. $2 + i$

In the complex plane, the point corresponding to $(1+3i)(3-i)$ is located in
A. the first quadrant
B. the second quadrant
C. the third quadrant

If the complex number $( a + i )( 1 - ai ) = 2$ , then $a =$
A. $-1$
B. $0$
C. $1$
D. $2$

Given $z = - 1 - \mathrm { i }$, then $| z | =$
A. 0
B. 1
C. $\sqrt { 2 }$
D. 2

Given $\frac { Z } { \mathrm { i } } = \mathrm { i } - 1$, then $Z =$ \_\_\_\_